Lecture 9 Graph Traversal - KOCWcontents.kocw.net/KOCW/document/2015/sungkyunkwan/suhuiseon… · SWE2004: Principles in Programming | Spring 2014 | Euiseong Seo ([email protected])

1 SWE2004: Principles in Programming | Spring 2014 | Euiseong Seo ([email protected])

Lecture 9 Graph Traversal

Euiseong Seo

([email protected])


Need for Graphs

One of unifying themes of computer science

Closely related to many daily life problems

• Navigation

• Circuit generation

• Social network services

• Games

• Computer networks

How can a problem be represented as a graph?

How to solve a graph problem?


Graph Notation

A graph G = (V, E)

• V is a set of vertices(nodes)

• E is a set of edges – E = (x, y) where x, y V

– Ordered or unordered pairs of vertices from V

Modeling of problems – What are vertices and edges in the followings? • Road networks

• Human interactions

• Program analysis


Flavors of Graphs

Undirected or directed

• A graph is undirected if edge (x, y) E implies that (y,x) E, too

• Otherwise, the graph is directed

w x

z y

w x

z y


Flavors of Graphs

Weighted or unweighted

• If each edge of a graph is assigned a numerical value, or weight, the graph is a weighted graph

• Otherwise, it is a unweighted graph

w x

z y

5

3

1

7

w x

z y

1

1

1

1


Flavors of Graphs

Cyclic or acyclic

• An acyclic graph does not contain any cycles

• Trees are connected acyclic undirected graphs

• Directed acyclic graphs are called DAGs


Data Structures for Graphs

Assume that a graph G = (V, E) contains n vertices and m edges

Adjacency matrix

• Use a n x n matrix M

• M[i,j] = 1, if (i,j) E

• M[i,j] = 0, if (i,j) E

• Pros – Easy to add or remove edges

– Easy to find a specific edge, (i,j), if exists

• Cons – Waste of memory space for sparse graphs



Adjacency lists in lists

1 2

5 4

3

1 2

5 4

3

2 5 / 1

1 4 / 2 5 3

2 4 / 3

2 3 / 4 5

4 2 / 5 1

2 5 / 1

5 / 2

3 4 / 3

4 / 5



Adjacency lists in matrices

• Use arrays instead of linked lists

• Looks like it combines the worst properties of both, but.

1 2

5 4

3

1 2 5 1 2

2 1 5 1 2

4 3 3 4

3 2 4 1 2

4 2 5 1 2

3 3

5 1 2 1 2

4 3

1 2

5 4

3

1 2 5 1 2

2 5 1

3 3 4 1 2

5 4 1


List in Array Representation

An undirected edge (x,y) appears twice, once as y in x’s list and once as x in y’s list


Tree Traversal

BFS (Breadth First Search)

DFS (Depth First Search)


BFS

s

2

5

4

7

8

3 6 9


s

2

5

4

7

8

3 6 9

0

Undiscovered

Discovered

Finished

Queue: s

Top of queue

2 1

Shortest path

from s

2


s

2

5

4

7

8

3 6 9

0

Queue: s 2

3

1

1

Undiscovered

Discovered

Finished

Top of queue

3


s

2

5

4

7

8

3 6 9

0

Queue: s 2 3

5

1

1

1

Undiscovered

Discovered

Finished

Top of queue

5


s

2

5

4

7

8

3 6 9

0

Queue: 2 3 5

4 1

1

1

2

Undiscovered

Discovered

Finished

Top of queue

4


s

2

5

4

7

8

3 6 9

0

Queue: 2 3 5 4

1

1

1

2

5 already discovered:

don't enqueue

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 2 3 5 4

1

1

1

2

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 3 5 4

1

1

1

2

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 3 5 4

1

1

1

2

6

2

Undiscovered

Discovered

Finished

Top of queue

6


s

2

5

4

7

8

3 6 9

0

Queue: 5 4 6

1

1

1

2

2

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 4 6

1

1

1

2

2

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 4 6

1

1

1

2

2

8 3

Undiscovered

Discovered

Finished

Top of queue

8


s

2

5

4

7

8

3 6 9

0

Queue: 6 8

1

1

1

2

2

3

7

3

Undiscovered

Discovered

Finished

Top of queue

7


s

2

5

4

7

8

3 6 9

0

Queue: 6 8 7

1

1

1

2

2

3

9

3

3

Undiscovered

Discovered

Finished

Top of queue

9


s

2

5

4

7

8

3 6 9

0

Queue: 8 7 9

1

1

1

2

2

3

9

3

3

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 7 9

1

1

1

2

2

3

3

3

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 7 9

1

1

1

2

2

3

3

3

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 7 9

1

1

1

2

2

3

3

3

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 9

1

1

1

2

2

3

3

3

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue: 9

1

1

1

2

2

3

3

3

Undiscovered

Discovered

Finished

Top of queue


s

2

5

4

7

8

3 6 9

0

Queue:

1

1

1

2

2

3

3

3

Undiscovered

Discovered

Finished

Top of queue Since Queue is empty, STOP!


DFS

Similar to Backtracking • Go as deep as you can • Next one is your siblings

Stack is an ideal candidate DFS(G, v)

for all edges e incident on v

do if edge e is unexplored then

w opposite(v, e) // return the end point of e distant to v

if vertex w is unexplored then

mark e as a discovered edge

recursively call DFS(G, w)

else

mark e as a back edge


Finding Paths

BFS Tree from x is unique

Parent[i] is the node that discovered node i during the BFS originated from x

Finding the shortest path from x to y in a undirected graph

• By following the chain of ancestors backward from y to the root


Connected Components

Many seemingly complicated problems reduce to

finding connected components

• 15-Puzzle

Connected components can be found by using repetitive application of DFS or BFS


Topological Sorting

One of the fundamental operations on DAGs

Construct an ordering of the vertices such that all directed edges go from left to right

• Cannot exist over cyclic graphs

This gives us a way to process each vertex before any of its successors

• Suppose we seek the shortest (or longest) path from x to y

• No vertex appearing after y in the topological order can contribute to any such path


Topological Sorting Algorithm

Definition

• A topological sort of a DAG G is a linear ordering of all its vertices such that if G contains a link (u,v), then node u appears before node v in the ordering

b

c a

d

b

c a

d

1

2 3

4



find source nodes (indegree = 0)

• if there is no such node, the graph is NOT DAG

c

a

b

e

d

f

in_deg=1

in_deg=0

in_deg=2

in_deg=1

in_deg=3

in_deg=1 Queue

Sorted: -

c



span c; decrement in_deg of a, b, e

• store a in Queue since in_deg becomes 0

c

a

b

e

d

f

in_deg=0

in_deg=0

in_deg=1

in_deg=1

in_deg=2

in_deg=1 Queue

Sorted: c

c

a


span a; decrement in_deg of b, f

• store b, f in Queue since ...

c

a

b

e

d

f

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=2

in_deg=1 Queue

Sorted: c a

a

b

f



span b; store d in Queue

c

a

b

e

d

f

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=2

in_deg=0 Queue

Sorted: c a b

b

f

d



span f; decrement in_deg of e • no node with in_deg = 0 is found

c

a

b

e

d

f

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=1

in_deg=0 Queue

Sorted: c a b f

f

d



span d; store e in Queue.

c

a

b

e

d

f

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=0 Queue

Sorted: c a b f d

d

e



span e; Queue is empty

c

a

b

e

d

f

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=0

in_deg=0 Queue

Sorted: c a b f d e

e


Documents

Lecture 9 Graph Traversal - KOCWcontents.kocw.net/KOCW/document/2015/sungkyunkwan/suhuiseon… · SWE2004: Principles in Programming | Spring 2014 | Euiseong Seo ([email protected])